Elementary classes of finite or finitely generated members of a semigroup variety

Under the printed definitions, the finite members of a semigroup variety form an elementary class exactly when the variety is trivial; the same is true of the finitely generated members.  A nontrivial variety contains a nontrivial finite semigroup whose powers have unbounded size, and compactness then produces forbidden infinite or uncountable models.

This is an additional literal-formulation note and is not included among the 18 independently audited counted results. No priority over the relevant classical literature is asserted.

AI disclosure: This problem was solved using GPT 5.6 Sol.

Mathematics subject: math.LO; source problem: OP-0063.